The Triangle Inequality Theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the third side.
Let's denote the lengths of the two sides as \(a = 3.7\) and \(b = 8.2\). Let \(c\) be the length of the third side. According to the Triangle Inequality Theorem, the following conditions must hold:
- \(a + b > c\)
- \(a + c > b\)
- \(b + c > a\)
Now we can apply these inequalities:
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From \(a + b > c\): \[ 3.7 + 8.2 > c \implies 11.9 > c \implies c < 11.9 \]
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From \(a + c > b\): \[ 3.7 + c > 8.2 \implies c > 8.2 - 3.7 \implies c > 4.5 \]
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From \(b + c > a\): \[ 8.2 + c > 3.7 \implies c > 3.7 - 8.2 \implies c > -4.5 \] (This inequality is always true since \(c\) is positive.)
Combining the inequalities from the first two conditions, we have: \[ 4.5 < c < 11.9 \]
Now we look at the responses provided (5.5, 5.5, 4.5, 4.5, 3.5, 3.5, 4.2). Based on our analysis, the possible measures of the third side must be greater than 4.5 and less than 11.9.
The only response that satisfies \(c > 4.5\) is 5.5 and possibly also the second 5.5 (since it's repeated).
Final answer: 5.5
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